Extension of possible symmetry groups One reason that physicists explored supersymmetry is because it offers an extension to the more familiar symmetries of quantum field theory. These symmetries are grouped into the
Poincaré group and internal symmetries and the
Coleman–Mandula theorem showed that under certain assumptions, the symmetries of the
S-matrix must be a direct product of the Poincaré group with a
compact internal symmetry group or if there is not any
mass gap, the
conformal group with a compact internal symmetry group. In 1971 Golfand and Likhtman were the first to show that the Poincaré algebra can be extended through introduction of four anticommuting spinor generators (in four dimensions), which later became known as supercharges. In 1975, the
Haag–Łopuszański–Sohnius theorem analyzed all possible superalgebras in the general form, including those with an extended number of the supergenerators and
central charges. This extended super-Poincaré algebra paved the way for obtaining a very large and important class of supersymmetric field theories.
Supersymmetry algebra Traditional symmetries of physics are generated by objects that transform by the
tensor representations of the
Poincaré group and internal symmetries. Supersymmetries, however, are generated by objects that transform by the
spin representations. According to the
spin-statistics theorem, bosonic fields
commute while fermionic fields
anticommute. Combining the two kinds of fields into a single
algebra requires the introduction of a
Z2-grading under which the bosons are the even elements and the fermions are the odd elements. Such an algebra is called a
Lie superalgebra. The simplest supersymmetric extension of the
Poincaré algebra is the
Super-Poincaré algebra. Expressed in terms of two
Weyl spinors, has the following
anti-commutation relation: :\{ Q_{ \alpha }, \bar {Q}_{ \dot{ \beta }} \} = 2( \sigma^{\mu} )_{ \alpha \dot{ \beta }} P_{\mu} and all other anti-commutation relations between the Qs and commutation relations between the Qs and Ps vanish. In the above expression P_\mu = -i \partial_\mu are the generators of translation and \sigma^\mu are the
Pauli matrices. There are
representations of a Lie superalgebra that are analogous to representations of a Lie algebra. Each Lie algebra has an associated
Lie group and a Lie superalgebra can sometimes be extended into representations of a
Lie supergroup.
Supersymmetric quantum mechanics Supersymmetric quantum mechanics adds the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. Supersymmetric quantum mechanics often becomes relevant when studying the dynamics of supersymmetric
solitons, and due to the simplified nature of having fields which are only functions of time (rather than space-time), a great deal of progress has been made in this subject and it is now studied in its own right. SUSY quantum mechanics involves pairs of
Hamiltonians which share a particular mathematical relationship, which are called
partner Hamiltonians. (The
potential energy terms which occur in the Hamiltonians are then known as
partner potentials.) An introductory theorem shows that for every
eigenstate of one Hamiltonian, its partner Hamiltonian has a corresponding eigenstate with the same energy. This fact can be exploited to deduce many properties of the eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory's fermions. Each boson would have a fermionic partner of equal energy.
Supersymmetry in quantum field theory In quantum field theory, supersymmetry is motivated by solutions to several theoretical problems, for generally providing many desirable mathematical properties, and for ensuring sensible behavior at high energies. Supersymmetric quantum field theory is often much easier to analyze, as many more problems become mathematically tractable. When supersymmetry is imposed as a
local symmetry, Einstein's theory of
general relativity is included automatically, and the result is said to be a theory of
supergravity. Another theoretically appealing property of supersymmetry is that it offers the only "loophole" to the
Coleman–Mandula theorem, which prohibits spacetime and internal
symmetries from being combined in any nontrivial way, for quantum field theories with very general assumptions. The
Haag–Łopuszański–Sohnius theorem demonstrates that supersymmetry is the only way spacetime and internal symmetries can be combined consistently. While supersymmetry has not been discovered at
high energy, see Section
Supersymmetry in particle physics, supersymmetry was found to be effectively realized at the intermediate energy of
hadronic physics where
baryons and
mesons are superpartners. An exception is the
pion that appears as a zero mode in the mass spectrum and thus protected by the supersymmetry: It has no baryonic partner. In 2021, a group of researchers showed that, in theory, N=(0,1) SUSY could be realised at the edge of a Moore–Read
quantum Hall state. However, to date, no experiments have been done yet to realise it at an edge of a Moore–Read state. In 2022, a different group of researchers created a computer simulation of atoms in 1 dimensions that had supersymmetric
topological quasiparticles.
Supersymmetry in optics In 2013,
integrated optics was found to provide a fertile ground on which certain ramifications of SUSY can be explored in readily-accessible laboratory settings. Making use of the analogous mathematical structure of the quantum-mechanical
Schrödinger equation and the
wave equation governing the evolution of light in one-dimensional settings, one may interpret the
refractive index distribution of a structure as a potential landscape in which optical wave packets propagate. In this manner, a new class of functional optical structures with possible applications in
phase matching, mode conversion and
space-division multiplexing becomes possible. SUSY transformations have been also proposed as a way to address inverse scattering problems in optics and as a one-dimensional
transformation optics.
Supersymmetry in dynamical systems All stochastic (partial) differential equations, the models for all types of continuous time dynamical systems, possess topological supersymmetry. In the operator representation of stochastic evolution, the topological supersymmetry is the
exterior derivative which is commutative with the stochastic evolution operator defined as the stochastically averaged
pullback induced on
differential forms by SDE-defined
diffeomorphisms of the
phase space. The topological sector of the so-emerging
supersymmetric theory of stochastic dynamics can be recognized as the
Witten-type topological field theory. The meaning of the topological supersymmetry in dynamical systems is the preservation of the phase space continuity—infinitely close points will remain close during continuous time evolution even in the presence of noise. When the topological supersymmetry is broken spontaneously, this property is violated in the limit of the infinitely long temporal evolution and the model can be said to exhibit (the stochastic generalization of) the
butterfly effect. From a more general perspective, spontaneous breakdown of the topological supersymmetry is the theoretical essence of the ubiquitous dynamical phenomenon variously known as
chaos,
turbulence,
self-organized criticality etc. The
Goldstone theorem explains the associated emergence of the long-range dynamical behavior that manifests itself as
noise,
butterfly effect, and the scale-free statistics of sudden (instantonic) processes, such as earthquakes, neuroavalanches, and solar flares, known as the
Zipf's law and the
Richter scale.
In finance In 2021, supersymmetric quantum mechanics was applied to
option pricing and the analysis of
markets in
finance, and to
financial networks.
Supersymmetry in mathematics SUSY is also sometimes studied mathematically for its intrinsic properties. This is because it describes complex fields satisfying a property known as
holomorphy, which allows holomorphic quantities to be exactly computed. This makes supersymmetric models useful "
toy models" of more realistic theories. A prime example of this has been the demonstration of S-duality in four-dimensional gauge theories that interchanges particles and
monopoles. The proof of the
Atiyah–Singer index theorem is much simplified by the use of supersymmetric quantum mechanics.
Supersymmetry in string theory Supersymmetry is an integral part of
string theory, a possible
theory of everything. There are two types of string theory, supersymmetric string theory or
superstring theory, and non-supersymmetric string theory. By definition of superstring theory, supersymmetry is required in superstring theory at some level. However, even in non-supersymmetric string theory, a type of supersymmetry called
misaligned supersymmetry is still required in the theory in order to ensure no physical
tachyons appear. Any string theories without some kind of supersymmetry, such as
bosonic string theory and the E_7 \times E_7, SU(16), and E_8
heterotic string theories, will have a tachyon and therefore the
spacetime vacuum itself would be unstable and would decay into some tachyon-free string theory usually in a lower spacetime dimension. There is no experimental evidence that either supersymmetry or misaligned supersymmetry holds in our universe, and many physicists have moved on from supersymmetry and string theory entirely due to the non-detection of supersymmetry at the LHC. Despite the null results for supersymmetry at the LHC so far, some
particle physicists have nevertheless moved to string theory in order to resolve the
naturalness crisis for certain supersymmetric extensions of the Standard Model. According to the particle physicists, there exists a concept of "stringy naturalness" in
string theory, where the
string theory landscape could have a power law statistical pull on soft SUSY breaking terms to large values (depending on the number of hidden sector SUSY breaking fields contributing to the soft terms). If this is coupled with an anthropic requirement that contributions to the weak scale not exceed a factor between 2 and 5 from its measured value (as argued by Agrawal et al.), then the Higgs mass is pulled up to the vicinity of 125 GeV while most sparticles are pulled to values beyond the current reach of LHC. (The Higgs was determined to have a mass of 125 GeV ±0.15 GeV in 2022.) An exception occurs for
higgsinos which gain mass not from SUSY breaking but rather from whatever mechanism solves the SUSY mu problem. Light higgsino pair production in association with hard initial state jet radiation leads to a soft opposite-sign dilepton plus jet plus missing transverse energy signal. == Supersymmetry in particle physics ==